NemoMultivariatePolynomial(R, VarSet, InternalOrdering)

jnpoly.spad line 195 [edit on github]

• R: NemoRing

• VarSet: List Symbol

• InternalOrdering: JuliaSymbol

This type is a basic representation of sparse, distributed multivariate polynomials using the Julia Nemo package. It is parameterized by the coefficient ring. The coefficient ring may be non-commutative, but the variables are assumed to commute. The monomial ordering used for internal storage can be one of :lex, :deglex or :degrevlex. For example: VarSet: List Symbol:=[x,y,z] V := OrderedVariableList(VarSet) -- eventually, for later use -- See SparseMultivariatePolynomial() -- E := IndexedExponents V PRing := NMP(NINT,VarSet, “lex”) x := x::V::PRing y := y::V::PRing z := z::V::PRing p:=x*2+3*y^2+17*z^13 p^7

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, Fraction Integer) -> % if R has Algebra Fraction Integer

*: (%, Integer) -> % if R has LinearlyExplicitOver Integer

from RightModule Integer

*: (%, NonNegativeInteger) -> %

*: (%, PositiveInteger) -> %

*: (%, R) -> %

from RightModule R

*: (Fraction Integer, %) -> % if R has Algebra Fraction Integer

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

+: (%, %) -> %

from AbelianSemiGroup

+: (%, Fraction Integer) -> % if R has QuotientFieldCategory NemoInteger

+: (Fraction Integer, %) -> % if R has QuotientFieldCategory NemoInteger

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, R) -> % if R has Field

from AbelianMonoidRing(R, IndexedExponents OrderedVariableList VarSet)

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

associates?: (%, %) -> Boolean if R has EntireRing

from EntireRing

associator: (%, %, %) -> %

from NonAssociativeRng

binomThmExpt: (%, %, NonNegativeInteger) -> % if % has CommutativeRing

from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList VarSet)

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

charthRoot: % -> Union(%, failed) if % has CharacteristicNonZero and R has PolynomialFactorizationExplicit or R has CharacteristicNonZero

from PolynomialFactorizationExplicit

coefficient: (%, IndexedExponents OrderedVariableList VarSet) -> R

from AbelianMonoidRing(R, IndexedExponents OrderedVariableList VarSet)

coefficient: (%, List OrderedVariableList VarSet, List NonNegativeInteger) -> %

from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

coefficient: (%, OrderedVariableList VarSet, NonNegativeInteger) -> %

from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

coefficients: % -> List R

from FreeModuleCategory(R, IndexedExponents OrderedVariableList VarSet)

coerce: % -> % if R has CommutativeRing

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> % if R has Algebra Fraction Integer or R has RetractableTo Fraction Integer

from Algebra Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

coerce: OrderedVariableList VarSet -> %

from CoercibleFrom OrderedVariableList VarSet

coerce: R -> %

from Algebra R

commutator: (%, %) -> %

from NonAssociativeRng

conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero and R has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

construct: List Record(k: IndexedExponents OrderedVariableList VarSet, c: R) -> %

from IndexedProductCategory(R, IndexedExponents OrderedVariableList VarSet)

constructOrdered: List Record(k: IndexedExponents OrderedVariableList VarSet, c: R) -> %

from IndexedProductCategory(R, IndexedExponents OrderedVariableList VarSet)

content: % -> R if R has GcdDomain

from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList VarSet)

content: (%, OrderedVariableList VarSet) -> % if R has GcdDomain

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

convert: % -> InputForm if R has ConvertibleTo InputForm

from ConvertibleTo InputForm

convert: % -> Pattern Float if R has ConvertibleTo Pattern Float

from ConvertibleTo Pattern Float

convert: % -> Pattern Integer if R has ConvertibleTo Pattern Integer

from ConvertibleTo Pattern Integer

convert: % -> String

from ConvertibleTo String

D: (%, List OrderedVariableList VarSet) -> %

from PartialDifferentialRing OrderedVariableList VarSet

D: (%, List OrderedVariableList VarSet, List NonNegativeInteger) -> %

from PartialDifferentialRing OrderedVariableList VarSet

D: (%, OrderedVariableList VarSet) -> %

from PartialDifferentialRing OrderedVariableList VarSet

D: (%, OrderedVariableList VarSet, NonNegativeInteger) -> %

from PartialDifferentialRing OrderedVariableList VarSet

degree: % -> IndexedExponents OrderedVariableList VarSet

from AbelianMonoidRing(R, IndexedExponents OrderedVariableList VarSet)

degree: (%, List OrderedVariableList VarSet) -> List NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

degree: (%, OrderedVariableList VarSet) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

differentiate: (%, List OrderedVariableList VarSet) -> %

from PartialDifferentialRing OrderedVariableList VarSet

differentiate: (%, List OrderedVariableList VarSet, List NonNegativeInteger) -> %

from PartialDifferentialRing OrderedVariableList VarSet

differentiate: (%, OrderedVariableList VarSet) -> %

from PartialDifferentialRing OrderedVariableList VarSet

differentiate: (%, OrderedVariableList VarSet, NonNegativeInteger) -> %

from PartialDifferentialRing OrderedVariableList VarSet

discriminant: (%, OrderedVariableList VarSet) -> % if R has CommutativeRing

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

eval: (%, %, %) -> %

from InnerEvalable(%, %)

eval: (%, Equation %) -> %

from Evalable %

eval: (%, List %, List %) -> %

from InnerEvalable(%, %)

eval: (%, List Equation %) -> %

from Evalable %

eval: (%, List OrderedVariableList VarSet, List %) -> %

from InnerEvalable(OrderedVariableList VarSet, %)

eval: (%, List OrderedVariableList VarSet, List R) -> %

from InnerEvalable(OrderedVariableList VarSet, R)

eval: (%, OrderedVariableList VarSet, %) -> %

from InnerEvalable(OrderedVariableList VarSet, %)

eval: (%, OrderedVariableList VarSet, R) -> %

from InnerEvalable(OrderedVariableList VarSet, R)

exquo: (%, %) -> Union(%, failed) if R has EntireRing

from EntireRing

exquo: (%, R) -> Union(%, failed) if R has EntireRing

from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList VarSet)

factor: % -> Factored % if R has PolynomialFactorizationExplicit

from UniqueFactorizationDomain

factor: % -> NemoFactored %

factor(p) returns the factorization of p.

factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

fmecg: (%, IndexedExponents OrderedVariableList VarSet, R, %) -> %

from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList VarSet)

gcd: (%, %) -> % if R has GcdDomain

from GcdDomain

gcd: List % -> % if R has GcdDomain

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has GcdDomain

from PolynomialFactorizationExplicit

ground?: % -> Boolean

from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList VarSet)

ground: % -> R

from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList VarSet)

hash: % -> SingleInteger if R has Hashable

from Hashable

hashUpdate!: (HashState, %) -> HashState if R has Hashable

from Hashable

isExpt: % -> Union(Record(var: OrderedVariableList VarSet, exponent: NonNegativeInteger), failed)

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

isPlus: % -> Union(List %, failed)

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

isTimes: % -> Union(List %, failed)

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

jlAbout: % -> Void

from JuliaObjectType

jlApply: (String, %) -> %

from JuliaObjectType

jlApply: (String, %, %) -> %

from JuliaObjectType

jlApply: (String, %, %, %) -> %

from JuliaObjectType

jlApply: (String, %, %, %, %) -> %

from JuliaObjectType

jlApply: (String, %, %, %, %, %) -> %

from JuliaObjectType

jlApply: (String, %, %, %, %, %, %) -> %

from JuliaObjectType

jlId: % -> String

from JuliaObjectType

jlRef: % -> SExpression

from JuliaObjectType

jlref: String -> %

from JuliaObjectType

jlType: % -> String

from JuliaObjectType

jmp2nmp: MultivariatePolynomial(VarSet, R) -> %

jnmp(p) converts the multivariate polynomial p to a Nemo multivariate polynomial.

latex: % -> String

from SetCategory

lcm: (%, %) -> % if R has GcdDomain

from GcdDomain

lcm: List % -> % if R has GcdDomain

from GcdDomain

lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if R has GcdDomain

from LeftOreRing

leadingCoefficient: % -> R

from IndexedProductCategory(R, IndexedExponents OrderedVariableList VarSet)

leadingMonomial: % -> %

from IndexedProductCategory(R, IndexedExponents OrderedVariableList VarSet)

leadingSupport: % -> IndexedExponents OrderedVariableList VarSet

from IndexedProductCategory(R, IndexedExponents OrderedVariableList VarSet)

leadingTerm: % -> Record(k: IndexedExponents OrderedVariableList VarSet, c: R)

from IndexedProductCategory(R, IndexedExponents OrderedVariableList VarSet)

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

linearExtend: (IndexedExponents OrderedVariableList VarSet -> R, %) -> R if R has CommutativeRing

from FreeModuleCategory(R, IndexedExponents OrderedVariableList VarSet)

listOfTerms: % -> List Record(k: IndexedExponents OrderedVariableList VarSet, c: R)

from IndexedDirectProductCategory(R, IndexedExponents OrderedVariableList VarSet)

mainVariable: % -> Union(OrderedVariableList VarSet, failed)

from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

map: (R -> R, %) -> %

from IndexedProductCategory(R, IndexedExponents OrderedVariableList VarSet)

mapExponents: (IndexedExponents OrderedVariableList VarSet -> IndexedExponents OrderedVariableList VarSet, %) -> %

from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList VarSet)

minimumDegree: % -> IndexedExponents OrderedVariableList VarSet

from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList VarSet)

minimumDegree: (%, List OrderedVariableList VarSet) -> List NonNegativeInteger

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

minimumDegree: (%, OrderedVariableList VarSet) -> NonNegativeInteger

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

monicDivide: (%, %, OrderedVariableList VarSet) -> Record(quotient: %, remainder: %)

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

monomial?: % -> Boolean

from IndexedProductCategory(R, IndexedExponents OrderedVariableList VarSet)

monomial: (%, List OrderedVariableList VarSet, List NonNegativeInteger) -> %

from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

monomial: (%, OrderedVariableList VarSet, NonNegativeInteger) -> %

from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

monomial: (R, IndexedExponents OrderedVariableList VarSet) -> %

from IndexedProductCategory(R, IndexedExponents OrderedVariableList VarSet)

monomials: % -> List %

from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

multivariate: (SparseUnivariatePolynomial %, OrderedVariableList VarSet) -> %

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

multivariate: (SparseUnivariatePolynomial R, OrderedVariableList VarSet) -> %

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

mutable?: % -> Boolean

from JuliaObjectType

nmp2smp: % -> SparseMultivariatePolynomial(Fraction Integer, OrderedVariableList VarSet) if R has QuotientFieldCategory IntegerNumberSystem

nmp2smp: % -> SparseMultivariatePolynomial(Integer, OrderedVariableList VarSet) if R hasn’t QuotientFieldCategory IntegerNumberSystem and R has IntegerNumberSystem

nothing?: % -> Boolean

from JuliaObjectType

numberOfMonomials: % -> NonNegativeInteger

from IndexedDirectProductCategory(R, IndexedExponents OrderedVariableList VarSet)

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if OrderedVariableList VarSet has PatternMatchable Float and R has PatternMatchable Float

from PatternMatchable Float

patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if OrderedVariableList VarSet has PatternMatchable Integer and R has PatternMatchable Integer

from PatternMatchable Integer

plenaryPower: (%, PositiveInteger) -> % if R has Algebra Fraction Integer or R has CommutativeRing

from NonAssociativeAlgebra %

pomopo!: (%, R, IndexedExponents OrderedVariableList VarSet, %) -> %

from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList VarSet)

prime?: % -> Boolean if R has PolynomialFactorizationExplicit

from UniqueFactorizationDomain

primitiveMonomials: % -> List %

from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

primitivePart: % -> % if R has GcdDomain

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

primitivePart: (%, OrderedVariableList VarSet) -> % if R has GcdDomain

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

recip: % -> Union(%, failed)

from MagmaWithUnit

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if R has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix R, vec: Vector R)

from LinearlyExplicitOver R

reducedSystem: Matrix % -> Matrix Integer if R has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: Matrix % -> Matrix R

from LinearlyExplicitOver R

reductum: % -> %

from IndexedProductCategory(R, IndexedExponents OrderedVariableList VarSet)

resultant: (%, %, OrderedVariableList VarSet) -> % if R has CommutativeRing

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

retract: % -> Fraction Integer if R has RetractableTo Fraction Integer

from RetractableTo Fraction Integer

retract: % -> Integer if R has RetractableTo Integer

from RetractableTo Integer

retract: % -> OrderedVariableList VarSet

from RetractableTo OrderedVariableList VarSet

retract: % -> R

from RetractableTo R

retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Fraction Integer

from RetractableTo Fraction Integer

retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer

from RetractableTo Integer

retractIfCan: % -> Union(OrderedVariableList VarSet, failed)

from RetractableTo OrderedVariableList VarSet

retractIfCan: % -> Union(R, failed)

from RetractableTo R

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

smaller?: (%, %) -> Boolean if R has Comparable

from Comparable

solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if R has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

squareFree: % -> Factored % if R has GcdDomain

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

squareFreePart: % -> % if R has GcdDomain

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

string: % -> String

from JuliaObjectType

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

support: % -> List IndexedExponents OrderedVariableList VarSet

from FreeModuleCategory(R, IndexedExponents OrderedVariableList VarSet)

totalDegree: % -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

totalDegree: (%, List OrderedVariableList VarSet) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

totalDegreeSorted: (%, List OrderedVariableList VarSet) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

unit?: % -> Boolean if R has EntireRing

from EntireRing

unitCanonical: % -> % if R has EntireRing

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %) if R has EntireRing

from EntireRing

univariate: % -> SparseUnivariatePolynomial R

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

univariate: (%, OrderedVariableList VarSet) -> SparseUnivariatePolynomial %

from PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

variables: % -> List OrderedVariableList VarSet

from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(R, IndexedExponents OrderedVariableList VarSet)

AbelianProductCategory R

AbelianSemiGroup

Algebra % if R has CommutativeRing

Algebra Fraction Integer if R has Algebra Fraction Integer

Algebra R if R has CommutativeRing

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer) if R has Algebra Fraction Integer

BiModule(R, R)

CancellationAbelianMonoid

canonicalUnitNormal if R has canonicalUnitNormal

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleFrom Fraction Integer if R has RetractableTo Fraction Integer

CoercibleFrom Integer if R has RetractableTo Integer

CoercibleFrom OrderedVariableList VarSet

CoercibleFrom R

CoercibleTo OutputForm

CommutativeRing if R has CommutativeRing

CommutativeStar if R has CommutativeRing

Comparable if R has Comparable

ConvertibleTo InputForm if R has ConvertibleTo InputForm

ConvertibleTo Pattern Float if R has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if R has ConvertibleTo Pattern Integer

ConvertibleTo String

EntireRing if R has EntireRing

Evalable %

FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList VarSet)

FreeModuleCategory(R, IndexedExponents OrderedVariableList VarSet)

FullyLinearlyExplicitOver R

FullyRetractableTo R

GcdDomain if R has GcdDomain

Hashable if R has Hashable

IndexedDirectProductCategory(R, IndexedExponents OrderedVariableList VarSet)

IndexedProductCategory(R, IndexedExponents OrderedVariableList VarSet)

InnerEvalable(%, %)

InnerEvalable(OrderedVariableList VarSet, %)

InnerEvalable(OrderedVariableList VarSet, R)

IntegralDomain if R has IntegralDomain

JuliaObjectRing

JuliaObjectType

JuliaRing

JuliaType

LeftModule %

LeftModule Fraction Integer if R has Algebra Fraction Integer

LeftModule R

LeftOreRing if R has GcdDomain

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer

LinearlyExplicitOver R

Magma

MagmaWithUnit

MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

Module % if R has CommutativeRing

Module Fraction Integer if R has Algebra Fraction Integer

Module R if R has CommutativeRing

Monoid

NemoRing

NemoType

NonAssociativeAlgebra % if R has CommutativeRing

NonAssociativeAlgebra Fraction Integer if R has Algebra Fraction Integer

NonAssociativeAlgebra R if R has CommutativeRing

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has EntireRing

PartialDifferentialRing OrderedVariableList VarSet

PatternMatchable Float if OrderedVariableList VarSet has PatternMatchable Float and R has PatternMatchable Float

PatternMatchable Integer if OrderedVariableList VarSet has PatternMatchable Integer and R has PatternMatchable Integer

PolynomialCategory(R, IndexedExponents OrderedVariableList VarSet, OrderedVariableList VarSet)

PolynomialFactorizationExplicit if R has PolynomialFactorizationExplicit

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RetractableTo OrderedVariableList VarSet

RetractableTo R

RightModule %

RightModule Fraction Integer if R has Algebra Fraction Integer

RightModule Integer if R has LinearlyExplicitOver Integer

RightModule R

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if R has CommutativeRing

UniqueFactorizationDomain if R has PolynomialFactorizationExplicit

unitsKnown

VariablesCommuteWithCoefficients